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The Woronowicz definition of a differential calculus over an algebra consists of a pair $(\Omega,$d$)$, where $\Omega$ is an $A-A$-bimodule, and $$ \text{d}:A \to \Omega, $$ is a bimodule map, satisfying the Leibniz rule d$(ab) = d(a)b + a$d$(b)$, and the condition that $\Omega = \text{span} ${$ a \text{d} b | a,b \in A$}.

Now one of the most common situations in which to consider these objects is when $A$ is a Hopf algebra (Drinfeld--Jimbo function algebra to be exact.). In this setting, it is natural to consider $q$-deformed definitions of classical objects. So it is natural to ask if one should consider a $q$-deformed Leibniz rule, for example of the form d$(ab) = d(a)b + q^c a$d$(b)$, where $c$ is some power that might be shown to depend on some braided category construction. Does any know of papers where this idea is explored? I guess someone must have tried this before.

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Such things are called twisted Kähler differential forms. Searching for that keyword should give results. – Mariano Suárez-Alvarez Feb 23 '13 at 18:48
Do these twisted Kahler forms offer any solution to the problem of bicovariant calculi of classical dimension on SU(2)? I didn't see any reference to this in your paper with Karoubi. – Antonio Nogueria Feb 24 '13 at 13:11
The derivations you mention in your edit are known as skew-derivations, as twisted derivations, as α-derivations, and under several other names. They show up all over the place (an egregius appearence is in the definition of Ore extensions) and have been studied. You should ask a more focused question, as that will probably augment the chances of getting a useful answer! – Mariano Suárez-Alvarez Mar 29 '13 at 22:56

As Mariano was mentioning what you're referring to is a special case of twisted derivations. Since you seem to be interested in differential calculus than a possible reasonable appearance of twisted derivations in quantum group theory emerges if one consider twisted Hochschild homology and its cyclic version as well (e.g. for some further explanations on relations).

Twisted Hochschild and cyclic homology may play the role of substitute of differential forms and De Rham complex, and were computed in a bunch of examples (again just one citation to give a starting point

One of the most interesting features of this twisted homology is that while usual Hochschild homology of quantum groups and quantum spaces faces a "dimension drop" problem, i.e. the homological dimension of quantum spaces is lower than the classical ones, the twisted one does not have this problem, in computed case, and has the same dimension as the classical one.

A possible explanation of this fact lies in the connection between quantum algebr and Poisson geometry: the Poisson manifolds underlying quantum groups and spaces are, in general, not unimodular, so that Poisson homology and cohomology fail to be one dual to the other.

A part form the Poisson geometrical interpretation, the dimension issue itself, together with its relation with the "modular aspects" of the theory (at an operator algebraic level quantization of the Poisson modular class is the Tomita-Takesaki operator) induced some people to work on new definition of spectral triples relying on twisted Hochschild (cyclic) theory rather than the usual one. They are named twisted spectral triples and I guess you can easily find a lot of papers dealing with them in many specific examples of quantum groups.

Sorry of all this is somewhat generic; putting details would mean writing a review paper of a whole area of research, rapidly developing in last 10 years...

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At least the Drinfel'd-Jimbo quantum groups (dual to the coordinate rings) have a structure, that is usually explored by using so-called skew-derivations satisfying exactly the rule you name (see e.g. Heckenberger Lecture Notes). What exactly are you trying to do?

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